The performance of a code under the maximum-likelihood (ML) decoder highly depends on the weight enumerating function (WEF). However, how to compute efficiently the WEF of a polar code or a polarization-adjusted convolutional (PAC) code is still an open problem. For the design of stand-alone polar codes, we consider enumerating the number of minimum-weight non-zero codewords of the polar code. The block error rate (BLER) under the ML decoder can be approximated as a function of the minimum weight of non-zero codewords and its multiplicity. On the other hand, for the design of PAC codes, we consider enumerating the WEF averaged over the ensemble of random PAC codes. The ML-BLER upper bound can be represented as a function of the WEF. The bit-channel selection algorithms for polar codes and PAC codes are proposed, which take both utilization of the polarization effect and the ML decoding performance as selection criteria. Simulation results show that the proposed stand-alone polar codes are competitive when the block length gets larger. Also, simulation results show that the proposed PAC codes yield excellent performance for a wide range of code rates and block lengths and outperform the 5G polar codes.

Ultra-reliable low-latency communication (URLLC) is an important feature brought by 5G New Radio (NR). By using a block code, the low-latency requirement in general requires a short block length in order to reduce the latency of transmitting the entire code block. The ultra-reliable communication requires a low-rate code to achieve high decoding reliability even under low signal-to-noise ratios (SNRs). This paper proposes a new class of coding and modulation schemes, termed permutation-coded modulations, for low-rate and short-block-length applications, such as URLLC. We show that permutation-coded modulations under maximum-likelihood (ML) decoding have remarkable performance levels that achieve the dispersion bounds with normal approximation (NA) for short block lengths. However, their encoding and ML-decoding complexities are prohibitive if the number of message bits transmitted per code block increases. To reduce the encoding and decoding complexities, we propose a trapezoidal permutation-coded modulation scheme which can be decoded efficiently by successive cancellation list (SCL) decoders. We also show that the trapezoidal permutation-coded modulations under SCL decoding can achieve the dispersion bounds with NA for short block lengths.

Conventional design methods of polar-coded modulation schemes aim to minimize the block error rate (BLER) under successive cancellation (SC) decoding. However, codes designed by conventional methods are not competitive under successive cancellation list (SCL) decoding. This paper presents a new design method based on the BLER upper bound under maximum-likelihood (ML) decoding (ML-BLER upper bound). The ML-BLER upper bound depends on the weight enumerating function (WEF) of the polar-coded modulation scheme over the squared Euclidean distance. In this paper, the polar-coded modulation is randomized by the concept of interleaved polar (i-polar) codes, and the WEF averaged over the ensemble of the polar-coded modulation schemes can be derived. Three polar-coded modulation schemes are considered, i.e., the bit-interleaved polar-coded modulation with a single interleaver (BIPCM-SI), the bit-interleaved polar-coded modulation with multiple interleavers (BIPCM-MI), and the multi-level polar-coded modulation (MLPCM). A new bit channel selection algorithm for polar-coded modulation schemes is proposed, which takes the polarization effect and the ML-BLER upper bound as design criteria. Design examples show that, under SCL decoding, the polar-coded modulation schemes (without CRC) with the proposed channel selection algorithm outperform those with conventional algorithms and are competitive as compared to the state-of-the-art 5G LDPC codes.

Polar codes were shown to be weak at finite block lengths even under maximum-likelihood (ML) decoding. To enhance polar codes, CRC-aided polar codes were proposed. It has been shown that CRC-aided polar codes under the successive cancellation list (SCL) decoder outperform LDPC codes and turbo codes. Recently, interleaved polar (i-polar) codes were proposed. By using a new bit channel selection algorithm, i-polar codes without CRC were shown to outperform the 5G LDPC codes. However, most research on polar codes focuses on the binary case. In this paper, a new family of nonbinary i-polar codes with random constellation mapping is proposed. A recursive form for calculating the average weight enumerating function (WEF) of the nonbinary i-polar code is derived. With the random constellation mapping, the upper bound on the BLER under ML decoding depends on the WEF of the nonbinary i-polar code. Then a new channel selection algorithm is proposed based on the polarization effect and the BLER upper bound. The algorithm generates a sequence, termed the i-polar sequence (IPS). Simulation results show that the nonbinary i-polar codes with the IPS outperform the 5G LDPC codes.

Abstract In this paper, an impulsive noise estimation algorithm for generating bit log‐likelihood ratios (LLRs) for channel coded systems in impulsive noise environments is proposed. This approach is to design the LLR detector in the maximum‐likelihood (ML) sense, which requires the parameters of the impulsive noise. The expectation‐maximisation (EM) algorithm is utilised to estimate the parameters of the Bernoulli–Gaussian (B–G) impulsive noise model. The estimated parameters is then used to generate the bit LLRs for the soft‐input channel decoder. Simulation results show that over a wide range of impulsive noise power, the proposed algorithm approaches the optimal performance (with ideal estimation) even under Middleton class‐A (M‐CA) impulsive noise models.

By inserting interleavers between intermediate stages of the polar encoder, a new class of polar codes, termed interleaved polar (i-polar) codes, is proposed. By the uniform interleaver assumption, we derive the weight enumerating function (WEF) and the input-output weight enumerating function (IOWEF) averaged over the ensemble of i-polar codes. The average WEF can be used to calculate the upper bound on the average block error rate (BLER) of a code selected at random from the ensemble of i-polar codes. Also, we propose a concatenated coding scheme that employs P high rate codes as the outer code and Q i-polar codes as the inner code with an interleaver in between. The average WEF of the concatenated code is derived based on the uniform interleaver assumption. Simulation results show that BLER upper bounds can well predict BLER performance levels of the concatenated codes. The results show that the performance of the proposed concatenated code with P=Q=2 is better than that of the CRC-aided i-polar code with P=Q=1 of the same length and code rate at high signal-to-noise ratios (SNRs). Moreover, the proposed concatenated code allows multiple decoders to operate in parallel, which can reduce the decoding latency and hence is suitable for ultra-reliable low-latency communications (URLLC).

In this paper, the hybrid beamforming design for multiple-input multiple-output orthogonal frequency-division multiplexing systems is studied over the indoor millimeter wave (mm-wave) channels. Under practical hybrid beamforming constraints for multi-carrier systems and the study of sub-systems, we propose the sub-system SVD (SS) hybrid beamforming design. To alleviate the complexity problem of channel estimation, reduction of the amount of the required channel state information (CSI) is also studied based on the statistical properties of indoor mm-wave channels. The SS hybrid beamforming algorithm is then extended to the limited feedback SS (LSS) scheme in which the transmit beamformers are designed based on limited information fed back from the receiver. A fast codeword selection algorithm is also developed to reduce the search complexity of the LSS algorithm. The simulation results show that our SS algorithm achieves the performance level of traditional full-digital beamforming with low complexity and the LSS algorithm has better performance and lower complexity than the previous algorithm, even if the amount of CSI is greatly reduced.

It has been shown that, under belief-propagation (BP) decoding, random-coset GF(q) low-density parity-check (LDPC) codes and irregular repeat-accumulate (IRA) codes with q-ary nonuniform signal constellations approach the unrestricted Shannon limit. In previous works, extrinsic information transfer (EXIT) charts were employed in the design of random-coset GF(q) LDPC and IRA modulation codes. However, in the EXIT charts, there is no closed-form expression for check node decoder (CND) curves. The CND curves for random-coset GF(q) LDPC and IRA modulation codes rely on Monte Carlo simulations, resulting in a high design complexity. This study presents new design methods of random-coset GF(q) LDPC and IRA modulation codes based on the average zero-word probability. The average zero-word probability serves as a surrogate for LLR messages, just as the mutual information acts as a surrogate for LLR messages in EXIT charts. Based on the average zero-word probability, closed-form expressions of CND input-output relations are derived for random-coset GF(q) LDPC and IRA modulation codes. Simple convergence criteria for random-coset LDPC and IRA modulation codes are proposed. Based on the proposed convergence criteria, six codes are designed with nonuniform signal constellations. Simulation results show that the proposed codes have near-capacity performances and are comparable with those designed based on EXIT charts.

Using nonbinary low-density parity-check (LDPC) codes with random-coset mapping, Bennatan and Burshtein constructed bandwidth-efficient modulation codes with remarkable performance under belief propagation (BP) decoding. However, due to the random nature of LDPC codes, most of the good LDPC codes found in the literature do not have a simple encoding structure. Thus, the encoding complexity of those LDPC codes can be as high asO(N2), whereNis the codeword length. To reduce the encoding complexity, in this paper, nonbinary irregular repeat-accumulate (IRA) codes with time-varying characteristic and random-coset mapping are proposed for bandwidth-efficient modulation schemes. The time-varying characteristic and random-coset mapping result in both permutation-invariance and symmetry properties, respectively, in the densities of decoder messages. The permutation-invariance and symmetry properties of the proposed codes enable the approximations of densities of decoder messages using Gaussian distributions. Under the Gaussian approximation, extrinsic information transfer (EXIT) charts for nonbinary IRA codes are developed and several codes of different spectral efficiencies are designed based on EXIT charts. In addition, by proper selection of nonuniform signal constellations, the constructed codes are inherently capable of obtaining shaping gains, even without separate shaping codes. Simulation results indicate that the proposed codes not only have simple encoding schemes, but also have remarkable performance that is even better than that constructed using nonbinary LDPC codes.

In this paper, convolutional codes are studied for unequal error protection (UEP) from an algebraic theoretical viewpoint. We first show that for every convolutional code there exists at least one optimal generator matrix with respect to UEP. The UEP optimality of convolutional encoders is then combined with several algebraic properties, e.g., systematic, basic, canonical, and minimal, to establish the fundamentals of convolutional codes for UEP. In addition, a generic lower bound on the length of a UEP convolutional code is proposed. Good UEP codes with their lengths equal to the derived lower bound are obtained by computer search.

It has been shown that, under belief-propagation (BP) decoding, random-coset GF(q) low-density parity-check (LDPC) codes and irregular repeat-accumulate (IRA) codes with q-ary nonuniform signal constellations approach the unrestricted Shannon limit. Extrinsic information transfer (EXIT) charts are employed in the design of random-coset GF(q) LDPC and IRA modulation codes. However, in the EXIT charts of random-cost GF(q) LDPC and IRA modulation codes, there is no closed-form expression for check node decoder (CND) curves. The CND curves for random-coset GF(q) LDPC and IRA modulation codes rely on Monte Carlo simulations, resulting in high design complexity. This study presents new design methods for random-coset GF(q) LDPC and IRA modulation codes based on average zero-word probability. Average zero-word probability serves as a surrogate for LLR messages, just as mutual information acts as a surrogate for LLR messages in EXIT charts. Based on average zero-word probability, closed-form expressions of CND input-output relations are derived for random-coset GF(q) LDPC and IRA modulation codes. Simple convergent criteria for both random-coset LDPC and IRA modulation codes are proposed. Based on the proposed convergent criteria, six codes are designed with nonuniform signal constellations. Simulation results show that the proposed codes have near-capacity performance and are better than any codes employed by equiprobable uniformly-spaced signal constellations.

A class of low-density parity-check (LDPC) codes with a simple 2-state trellis structure is presented. For LDPC decoding, the conventional belief propagation (BP) algorithm consists of numerous sub-decoders of single-parity check codes and exchanges information between sub-decoders in an iterative manner. If the single-parity check codes can be constructed and grouped in a proper way, the decoder can be decomposed into few identical 2-state trellis decoders. Therefore, instead of numerous sub-decoders of single-parity check codes, an iterative decoding algorithm based on few sub-decoders over 2-state trellis is proposed. The proposed decoding algorithm improves the efficiency of message passing between sub-decoders and hence provides a fast convergent rate as compared to the standard BP algorithm. Simulation results show that the proposed scheme provides a better performance and a fast convergent rate as compared to those of standard BP algorithm. The result also shows that the proposed algorithm has a similar performance as that of asynchronous replica shuffled BP algorithm and has a slightly inferior performance than that of synchronous replica shuffled BP algorithm. However, complexity analysis shows that our proposed algorithm has complexity that is lower than that of the replica shuffled BP algorithm.

A new construction of good, easily encodable, and soft-decodable codes is proposed in this paper. The construction is based on serially concatenating several simple 1+D convolutional codes as the outer code, and a rate-1 1/(1+D) accumulate code as the inner code. These codes have very low encoding complexity and require only one shift-forward register for each encoding branch. The input-output weight enumerators of these codes are also derived. Divsalariquests simple bound technique is applied to analyze the bit error rate performance, and to assess the minimal required signal-to-noise ratio (SNR) for these codes to achieve reliable communication under AWGN channel. Simulation results show that the proposed codes can provide good performance under iterative decoding.

Due to the random nature of LDPC codes, most of the good LDPC codes found in the literature do not possess a simple encoding structure. Thus, the encoding complexity of those LDPC codes can be as high asO(N2), where N is the codeword length. To reduce the encoding complexity, in this paper, binary irregular repeat accumulate (IRA) codes are extended to the nonbinary cases for bandwidth-efficient modulation schemes. By proper selection of nonuniform signal constellations, the constructed codes are inherently capable to obtain shaping gains even without shaping codes. Under Gaussian approximation, extrinsic information transfer (EXIT) charts for nonbinary IRA codes are developed and several codes of different spectral efficiencies are designed based on EXIT charts. Simulation results indicate that the proposed codes not only have simple encoding schemes, but also have remarkable performances that are comparable to those of nonbinary LDPC codes.

Recently an algebraic construction of (nttimes Q) space-frequency (SF) codes over finite field Fqwas proposed for use in MIMO-OFDM systems, where nt is the number of transmit antenna and Q = qnt- 1 is the number of subcarriers employed in the code. One inconvenience arising from that construction is that the number of subcarriers Q can sometimes be insufficient for constructing codes of large minimum column distance. To completely eliminate this disadvantage, an alternative construction of SF codes with Q = qm- 1 is provided in this paper, whenever m is a multiple of nt. Lower bounds on the minimum rank and column distances of the proposed construction are also given. Simulation results show that the newly constructed codes provide a significant improvement in SNR compared to other SF codes available in the literature.

In this paper, we investigate a novel coding idea proposed previously to enable under-sampled receivers. An under-sampled receiver can sample the received baseband signals at only a fraction of the Nyquist rate and turns out to be an effective solution to the bottleneck of high-rate sampling and processing in ultra-wideband (UWB) communications. The spectrum aliasing problem can be solved by an analogy between an under-sampled system and a multiple-antenna one. However, underlying differences between the two systems exist and motivate the study of the fundamental limits of a coded under-sampled system. The study is carried out by characterizing the optimal coding structures with and without channel state information at transmitter (CSIT). A practical selective coding structure is also developed to provide satisfactory performance with reduced CSIT requirement. Finally, simulations are conducted to verify the theoretical characterization. Our results indicate that an under-sampled UWB system can benefit from power-saving, cost reduction, and full multipath diversity at the expanse of little or confined performance degradation

Orthogonal frequency division multiplexing (OFDM) has been proposed for use as the physical layer of ultra-wideband (UWB) systems for high-rate, short-range personal area networking (PAN). For ultra-wideband systems, there is a constraint on the maximum power spectral density for the transmitted signal. Therefore, the bandwidth of the transmitted spectrum must be spread widely by a bandwidth expansion scheme so that the transmitted power spectral density can be kept as low as possible. In this paper, frequency expansion of the UWB system is achieved by using a simple frequency-diversity coding scheme. A major issue for the frequency-diversity coding scheme is that the receiver must sample the baseband received signal using high-sampling-rate analog-to-digital converters (ADCs) for discrete signal processing (DSP). However, such high-sampling-rate ADCs and DSP are expensive and have high power consumption. One advantage of the proposed frequency-diversity coding scheme is that the sampling rate of the baseband ADCs and DSP can be less then the Nyquist rate. The aliasing phenomenon occurs due to the reduced sampling rate, yet it, however, appears as transmission diversity to the receiver. The performance of the frequency-diversity coded OFDM system with an under-sampling-rate receiver is analyzed by evaluating the pairwise error probability. From the analysis of the pairwise error probability, design criteria for the frequency-diversity coded OFDM are obtained. A practical construction of frequency-diversity codes is proposed based on linear block codes. Simulation and analytical results for frequency-diversity coded OFDM systems are presented. The results show that a significant diversity/coding gain can be achieved with the under-sampling-rate receiver.

Constructions of space-frequency (SF) codes for MIMO-OFDM systems with n/sub t/ transmit antennas and Q subcarriers are considered in this paper. Arising from the pairwise-error-probability analysis, in addition to the rank distance criterion, the minimum column distance of (n/sub t/ /spl times/ Q) SF codes serves as another benchmark in code design. Codes with larger minimum column distance are expected to have better performance. Following this observation, two code constructions are presented. The first construction is obtained by right-multiplying the code matrices in a maximal rank-distance (MRD) code by a fixed, (Q /spl times/ Q) nonsingular matrix. Codes obtained from this construction are called linearly transformed MRD (LT-MRD) codes in this paper. Minimum column distance of the LT-MRD codes, when averaged over all code ensembles, is shown to meet the Gilbert-Varshamov bound. The second construction is reminiscent of the construction of the Reed-Solomon codes except that the code polynomials are now selected according to the cyclotomic cosets of the underlying field. Exact minimum rank distances and bounds on the minimal column distance of these codes are presented.

In this paper, the reduction of the peak-to-average power ratio (PAPR) of M-quadrature amplitude modulation (QAM) signals in orthogonal frequency division multiplexing (OFDM) systems are studied. The block coded modulation (BCM) techniques and Golay sequences to trade off the PAPR, the code rate, and the squared Euclidean distance in M-QAM OFDM signals are combined. In particular, 16-QAM and 64-QAM OFDM sequences with low PAPR and good squared Euclidean distance are presented.

A new construction of direct current (DC)-free error-correcting codes based on convolutional codes is proposed. The new code is constructed by selecting a proper subcode from a convolutional code composed of two different component codes. The encoder employs a Viterbi algorithm as the codeword selector so that the selected code sequences satisfy the DC constraint. A lower bound on the free distance of such codes is proposed, and a procedure for obtaining this bound is presented. A sufficient condition for these codes to have a bounded running digital sum (RDS) is proposed. Under the assumption of a simplified codeword selection algorithm, we present an upper bound on the maximum absolute value of the RDS and derive the sum variance for a given code. A new construction of standard DC-free codes, i.e., DC-free codes without error-correcting capability, is also proposed. These codes have the property that the decoder can be implemented by simple symbol-by-symbol hard decisions. Finally, under the new construction, we propose several codes that are suitable for the systems that require small sum variance and good error-correction capability.

We consider a practical maximum-likelihood sequence estimation (MLSE) equalizer on multipath fading channels in conjunction with an adaptive channel estimator consisting of a least mean square (LMS) estimator and a linear channel predictor, instead of assuming perfect channel estimates. A new LMS estimator model is proposed which can accurately characterize the statistical behavior of the LMS estimator over multipath fading channels. Based on this model, a new upper-bound on block error rate is derived under the consideration of imperfect channel estimates. Computer simulations verify that our analytical results can correctly predict the real system performance and are applicable over a wide range of the step size parameter of the LMS estimator.

In the literature the performance of joint maximum-likelihood sequence estimation for trellis-coded modulation systems was analyzed under the assumption that fading is so slow that the channel does not change during all error events. In this paper we extend the performance analysis to general fading processes by considering the correlation function of the time-variant channel impulse response instead of assuming constant fading. An easily evaluated closed-form upper bound is derived for the pairwise error probability. The bit error rate is then estimated by using a truncated union bound. Computer simulations show that our analytical results are good for all cases considered especially when diversity reception is used.>

Commentson a paper by Rouanne and Costello Jr. (IEEE Trans. Inform. Theory, vol.34, p.1011-20, 1988). It is shown that the derivation in the above paper is incorrect and hence the lower bound obtained is not valid for general trellis-coded modulation schemes. A corrected bound is presented to remedy this flaw. Also shown is a similar lower bound derived from the average distance structure. Both bounds have similar asymptotic behavior and are linear in the constraint length. Unfortunately, they are bad and not very useful for short and medium constraint lengths. The present author points out that there is still much room for further research in finding a good lower bound on the maximal minimum distance for trellis-coded modulation schemes.>